Noncentrosymmetric metal electrodes for ferroic devices

ABSTRACT

A ferroelectric heterostructure may comprise a ferroelectric layer comprising a ferroelectric material and a first electrode layer comprising a first noncentrosymmetric metal, the first electrode layer disposed on the ferroelectric layer to form a ferroelectric-first electrode interface, wherein the ferroelectric layer is characterized by exhibiting an electric polarization and the first electrode layer is characterized by exhibiting polar ionic displacements and further wherein, a component of the polar ionic displacements of the first electrode layer is parallel to a component of the electric polarization of the ferroelectric layer.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. Provisional PatentApplication No. 62/371,345 that was filed Aug. 5, 2016, the entirecontents of which are hereby incorporated by reference.

REFERENCE TO GOVERNMENT RIGHTS

This invention was made with government support under W911NF-15-1-0017awarded by the Army Research Office. The government has certain rightsin the invention.

BACKGROUND

Ferroelectric perovskite oxides with a spontaneous electric polarizationhave potential applications in nonvolatile random access memories andhigh-density data storage devices. Following consumer demands for higherdensity data storage technologies, a continuing miniaturization ofelectronic components has been the major focus of researchers andindustry. To achieve this aim for devices which integrate ferroelectricoxides, one of the most important issues is how to maintainferroelectricity (and electric polarizations) when the component sizereduces to the nanometer scale. Specifically, finding effective androbust routes to reduce the critical thickness (before ferroelectricityvanishes) and thus improve ferroelectric stability of ultrathin films isof critical importance.

SUMMARY

Provided are ferroelectric heterostructures and devices including theheterostructures, e.g., capacitors and field-effect transistors.

In one aspect, ferroelectric heterostructures are provided. In anembodiment, a ferroelectric heterostructure comprises a ferroelectriclayer comprising a ferroelectric material and a first electrode layercomprising a first noncentrosymmetric metal, the first electrode layerdisposed on the ferroelectric layer to form a ferroelectric-firstelectrode interface, wherein the ferroelectric layer is characterized byexhibiting an electric polarization and the first electrode layer ischaracterized by exhibiting polar ionic displacements and furtherwherein, a component of the polar ionic displacements of the firstelectrode layer is parallel to a component of the electric polarizationof the ferroelectric layer.

The ferroelectric heterostructure may be incorporated into a capacitor.In an embodiment, a capacitor comprises the ferroelectricheterostructure described above and a second electrode layer comprisinga second noncentrosymmetric metal, wherein the ferroelectric layer isbetween the first and second electrode layers to further form aferroelectric-second electrode interface, wherein the second electrodelayer is characterized by exhibiting polar ionic displacements andfurther wherein, a component of the polar ionic displacements of thesecond electrode layer is parallel to the component of the electricpolarization of the ferroelectric layer.

Either ferroelectric heterostructure may be incorporated into afield-effect transistor. In an embodiment, a transistor comprises eitherferroelectric heterostructure disposed over a substrate, a sourceelectrically coupled to the substrate and a drain in electricallycoupled to the substrate.

Other principal features and advantages of the invention will becomeapparent to those skilled in the art upon review of the followingdrawings, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the invention will hereafter be describedwith reference to the accompanying drawings.

FIG. 1 shows the structure of a conventional ferroelectric capacitorincluding a BaTiO₃ film between SrRuO₃ electrodes that areshort-circuited.

FIG. 2 shows the evolution of the energy as a function of the soft-modedistortion for the capacitor of FIG. 1. These are the first-principlescalculations reported by J. Junquera et al., Nature 422, 506 (2003) fordifferent thicknesses of the ferroelectric film: m=2, m=4, m=8, andm=10.

FIG. 3 shows the structure of a ferroelectric capacitor according to anillustrative embodiment of the present disclosure. The capacitor makesuse of noncentrosymmetric metal (NSCM) electrodes, i.e., polar metalelectrodes. Such a ferroelectric capacitor may include a BaTiO₃ filmbetween LiOsO₃ electrodes that are short-circuited:[LiO—(OsO₂—LiO)₅/TiO₂—(BaO—TiO₂)_(m)]. Also shown is the atomicrepresentation of the related supercell thatwas simulated (for the casem=1). P_(NCSM) and P_(BaTiO3) are the polar displacements (ferroelectricdistortions) in LiOsO₃ and BaTiO₃, respectively.

FIGS. 4A-4D show a symmetric nanocapacitor according to an illustrativeembodiment of the present disclosure. FIG. 4A shows a centrosymmetricnanocapacitor with insulating NaNbO₃ (m=2) between LiOsO₃ electrodes(n=6). FIG. 4B shows the equilibrium structure of the ferroelectriccapacitor [LiO—(OsO₂—LiO)₆/NbO₂—(NaO—NbO₂)₂]. The direction of the polardisplacements in the electrodes and the ferroelectric film (A_(NNO)) areindicated with arrows. FIG. 4C shows a magnification of the LiO/NbO₂interface of the paraelectric aristotype. FIG. 4D shows a magnificationof the LiO/NbO₂ interface of the ground state structure. The FE behaviorat the interface is due to the Li, Nb, and O displacements. Thedirection of the polar displacements at the interfaces for theparaelectric and the ground state structure is indicated with arrows.

FIG. 5A depicts projected densities-of-states (DOS) for the NaNbO₃ layerin the LiOsO₃/NaNbO₃/LiOsO₃ (m=1) nanocapacitor within LDA. FIG. 5Bshows projected densities-of-states (DOS) for the NaNbO₃ layer in theLiOsO₃/NaNbO₃/LiOsO₃ (m=1) nanocapacitor within HSE06. The inset showsthe LiOsO₃/NaNbO₃/LiOsO₃ (m=1) nanocapacitor.

FIG. 6 depicts energetic gain with increasing polar mode amplitudeA_(NNO) in the NaNbO₃ film for different thicknesses: m=1 (circles), m=2(squares) and m=3 (diamonds). A finite value of A_(NNO) leads to anenergetic gain for all m, indicating the disappearance of the criticalthickness. Note that the energy of the polar ground state structure withA_(NNO)=0 is taken as reference for each value of m. The energy gainincreases as the thickness of the ferroelectric film increases owing tothe reduced effects from the depolarizing field, which dominates thinnerfilms.

FIGS. 7A and 7B depict typical behavior of a comparative conventionalnanoscale ferroelectric capacitor below the critical thickness. FIG. 7Ashows a nanocapacitor consisting of a BaTiO₃ film (m=2) betweencentrosymmetric SrRuO₃ (n=6) electrodes. The arrow indicates thedirection of the polar displacements in BaTiO₃. FIG. 7B depicts energyas a function of the polar mode amplitude A_(BTO) in BaTiO₃. The energyincreases for all thicknesses m analyzed.

FIGS. 8A and 8B show the energetic landscape of a BaTiO₃ nanocapacitorwith “polar” SrRuO₃ electrodes. Energetic gain is shown for differentdielectric thicknesses: m=1 (circles), m=2 (squares) and m=3 (diamonds).The Sr displacements given with respect to the centrosymmetric groundstate are fixed to 0.07 Å in FIG. 8A (see inset) and 0.14 Å in FIG. 8B.The energy of the polar structure with A_(BTO)=0 Å is taken as referencefor each m.

FIG. 9 shows the planar and macroscopically averaged electrostaticpotential of the BaTiO₃ ferrocapacitor (m=2) along [001] between polar(solid line) and centrosymmetric (broken line) SrRuO₃ electrodes. Thedifference, Δ, is also shown. The amplitude of the polar distortion inthe BaTiO₃ film is fixed to ˜0.6 Å, which corresponds to the minimum inFIG. 8B. The configuration with paraelectric BaTiO₃ is used asreference.

FIGS. 10A-10D show geometries of illustrative ferroelectric field-effecttransistors (FE-FETs). FIG. 10A shows a FE-FET including a metal gateelectrode, a ferroelectric layer, and a semiconductor substrate; FIG.10B shows a FE-FET including a metal gate electrode, a ferroelectriclayer, an insulator (buffer) layer, and a semiconductor substrate; FIG.10C shows a FE-FET including a metal gate electrode, a ferroelectriclayer, another layer of metal, an insulator (buffer) layer, and asemiconductor substrate; and FIG. 10D shows a FE-FET including a metalgate electrode, a ferroelectric layer, a conducting oxide layer, and aperovskite substrate. As further described below, the disclosednoncentrosymmetric metals may be used for the metal gateelectrodes/layer of metal (FIG. 10C)/conducting oxide (FIG. 10D) whilethe disclosed ferroelectric materials may be used for the ferroelectriclayers shown in these figures.

DETAILED DESCRIPTION

Provided are ferroelectric heterostructures and devices including theheterostructures, e.g., capacitors and field-effect transistors.

The present disclosure makes use of noncentrosymmetric metal electrodesto cause the critical thickness for ferroelectricity to effectivelyvanish (become non-existent) for a variety of ferroelectric materials.This unique approach solves the problem of miniaturization from anentirely different point of view as compared to conventional approaches.Specifically, the present approach focuses on the properties of themetal electrode of ferroelectric heterostructures. By contrast,conventional approaches make use of centrosymmetric metal electrodes andinstead focus on the properties of the ferroelectric layer of theheterostructures.

The miniaturization problem is illustrated with the conventionalferroelectric capacitor shown in FIG. 1. The capacitor includes aferroelectric BaTiO₃ (BTO) film between centrosymmetric SrRuO₃ (SRO)electrodes. Owing to an intrinsic size effect and a depolarizing fieldthat results from poor screening by the itinerant electrons in theelectrode, the thickness of ferroelectric film is limited to a criticalthickness. Below this critical thickness, the ferroelectric polarizationinevitably disappears, rendering the devices non-functional.

Indeed, first-principles calculations by show that a BTO film loses itsferroelectricity at a critical thickness of 2.4 nm. (J. Junquera, etal., Nature 422, 506 (2003).) These calculations, shown in FIG. 2,reveal that for different thicknesses of the BaTiO₃ layer, indicated bythe symbols, there is a clear evolution from a stable ferroelectric (atlarge thicknesses, m>6) to an unstable ferroelectric state (and smallthicknesses, m≤6). Note that the symbols in FIG. 2 show the evolution ofthe energy (with respect to the reference paraelectric phase) uponfreezing-in atomic displacements parameterized by ξ with increasingamplitude. Thus, the change in stability of the ferroelectric behavioroccurs at a critical thickness around m=6 (26 Å). For largerthicknesses, the energy decreases when the atoms follow the bulksoft-mode path, clearly demonstrating that the film has a ferroelectricground state (in agreement with experimental results for the samethickness (see Appl. Phys. Lett. 75, 856-858 (1999)). By contrast, belowthe critical thickness, the energy is minimized for the paraelectricconfiguration. These theoretical predictions were confirmedexperimentally (see Appl. Phys. Lett. 86, 102907 (2005). (Forconventional ferroelectric capacitor behavior, see also FIGS. 7A and 7B,further described in the Examples, below.)

Thus, as used in the present disclosure, the phrase “critical thicknessvalue” refers to the thickness value at which a ferroelectric layerloses its ferroelectricity. This critical thickness value will bedifferent for different ferroelectric layers made up of differentferroelectric materials. However, the critical thickness value can becalculated and/or measured using the techniques described above (as wellas those described in the Examples, below).

As noted above, by contrast to prior art ferroelectric heterostructures,the present ferroelectric heterostructures make use ofnoncentrosymmetric metal electrodes. In embodiments, a ferroelectricheterostructure comprises a ferroelectric layer comprising aferroelectric material and a first electrode layer comprising a firstnoncentrosymmetric metal, the first electrode layer disposed on theferroelectric layer. The ferroelectric layer and the first electrodelayer are in direct contact so as to form a ferroelectric-firstelectrode interface.

As demonstrated in detail in the Examples section, this approach ofusing noncentrosymmetric metal electrodes is capable of maintaining theferroelectricity of a variety of ferroelectric layers made of differentferroelectric materials even when the thickness of the ferroelectriclayer is on the (sub)-nanometer scale, including below the criticalthickness value of the ferroelectric layer. Such devices andferroelectric layers include capacitors and memories based on BaTiO₃,utilizing its high dielectric response; electromechanical transducersbased on Pb(Zr_(0.52)Ti_(0.48))O₃ (PZT), utilizing its highpiezoelectric coefficient; pyroelectrics such as PbTiO₃ and(Sr,Ba)Nb₂O₆; and electrooptical components based on LiNbO₃. Thisfinding that noncentrosymmetric metal electrodes can cause the criticalthickness value for ferroelectricity to effectively vanish (becomenon-existent) enables the aggressive scaling of ferroelectric capacitorsbelow current dimensional constraints. This, in turn, enables the nextgeneration of IT equipment such as ultra-high-speed mobile computing,communication devices, and sophisticated sensors.

By “noncentrosymmetric metals,” it is meant a distinct class ofcompounds characterized by the following properties. First,noncentrosymmetric metals are crystalline compounds characterized by acrystal structure that lacks inversion symmetry (i.e., has no inversioncenter). Regarding crystal structure, there are 21 noncentrosymmetricpoint groups: 1, 2, m, 222, mm2, 4, 422, 4, 4mm, 4 2m, 3, 32, 3m, 6,622, 6, 6 mm, 6 2m, 23, 432, and 4 3m. Second, the noncentrosymmetricmetals of the present disclosure are polar, characterized by a polaraxis, polar ionic displacements, and are characterized by one of thefollowing 10 polar point groups, which are a subset of theaforementioned 21 noncentrosymmetric point groups: 1, 2, m, mm2, 4, 4mm,3, 3m, 6, and 6 mm. Third, noncentrosymmetric metals exhibit metallicbehavior, i.e., an ability to conduct electrons with a minimumresistivity value of approximately 1000 μΩ·cm at room temperature. Thefree carriers may arise from either a high density of partially occupiedelectronic states at the Fermi level as in a band conductor or bydegenerately doping a ferroelectric material to achieve the requiredresistivity minimum. However, the doped and conducting ferroelectricmaterial must maintain its polar ionic displacements as described above.The present disclosure may refer to “noncentrosymmetric metals” as“polar metals” and “ferroelectric metals.” These three terms are usedinterchangeably in the present disclosure.

A variety of noncentrosymmetric metals may be used in the ferroelectricheterostructures. The first electrode layer may comprise combinations ofdifferent types of noncentrosymmetric metals. Illustrativenoncentrosymmetric metals include those listed in Table 1, below. Othernoncentrosymmetric metals include those disclosed in Nature Materials12, 1024-1027 (2013); Nat. Commun. 5, 3432 (2014); Phys. Rev. B90,094108 (2014); and J. Mater. Chem. C 4, 4000-4015 (2016), each of whichis hereby incorporated by reference in its entirety.

TABLE 1 Illustrative noncentrosymmetric metals. Rh₂Ga₉ La₂NiAl₇Sr₃Cu₈Sn₄ Ru₇B₃ CaAlSi Li₁₇Ag₃Sn₆ BiPd Li₂IrSi₃ LiOsO₃ UIr CeRuSi₃Ca₃Ru₃O₇ Mg₂Al₃ SrAuSi₃ RESr₂Cu₂GaO₇, wherein RE is any of La through Ybor Y Ir₉Al₂₈ CePt₃Si BaVS₃ Ir₂Ga₉ AIrSi₃ wherein A is Ca AV₆S₈, whereinA or Ce is K, Rb, Cs, or Tl Rh₂Ga₉ ARhSi₃, wherein A is Ce La₄Mg₅Ge₆ orLa γ-Bi₂Pt APtSi₃, wherein A is Ca, La₄Mg₇Ge₆ Ba, or EuAu_(6.05)Zn_(12.51) ACoGe₃, wherein A is Ce, Yb₂Ga₄Ge₆ Pr, or LaBa₂₁Al₄₀ LaASi₃, wherein A is Ir ErPdBi or Pd Cr₅Al₈ CeAGe₃, wherein Ais Ir LuPtBi or Rh Mn₅Al₈ EuNiGe₃ Ce₂Rh₃(Pb, Bi)₅ Cu_(7.8)Al₅ EuPdGe₃Eu₂Pt₃Sn₅ Cu₇Hg₆ LaAGe₃, wherein A is Co, Lu₄Zn₅Ge₆ Fe, Ir or Rh NbS₂SrAGe₃, wherein A is Pd IrMg_(2.03)In_(0.97) or Pt Sn₄As₃ REPdIn₂,wherein RE is IrMg_(2.20)In_(0.80) Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er,Tm, Lu or Y Sn₄P₃ REAuGe, wherein RE is CeAuSn Ce, Lu, Sc or Ho REPt₃B,wherein La₁₅Ge₉X, wherein X is LaGeSi₃ RE is La, Pr, or Nd C, Co, Fe, orNi La₅B₂C₆ HoAuGe LaNiC₂ CeCuSn

A variety of ferroelectric materials may be used in the ferroelectricheterostructures and the ferroelectric layer may comprise combinationsof different types of ferroelectric materials. Ferroelectric materialsalso make up a distinct class of compounds characterized byferroelectricity, i.e., reversible spontaneous electric polarization.

In embodiments, the ferroelectric material is an oxide such as aperovskite oxide. A variety of perovskite oxides may be used, includingcubic perovskite oxides and double perovskite oxides. Illustrativeferroelectric perovskite oxides include the following: BaTiO₃;Ba_(x)Sr_(1-x)TiO₃ wherein 1>x>0; PbTiO₃; BaZrO₃; Pb(Zr_(x)Ti_(1-x))O₃wherein 1≥x≥0; (Sr,Ba)Nb₂O₆; NaNbO₃; BiFeO₃; and YMnO₃. Otherferroelectric oxides include the Ruddlesden-Popper oxideSr_(x)Ca_(3-x)Ti₂O₇ wherein 1>x>0; the Aurivillius oxide SrBi₂Ta₂O₉; andthe lithium-niobate structured oxide LiNbO₃. Additional multicationferroelectric oxides include Sr(Ta,Nb)₂O₇; Gd₂(MoO₄)₃; and Pb₅Ge₃O₁₁.However, the ferroelectric material need not be an oxide. Otherillustrative ferroelectric materials that are non-oxides include BaMnF₄,GeTe, SrAlF₅, and SbSI.

Other ferroelectric materials include proper ferroelectrics, improperferroelectrics, hybrid improper ferroelectrics, and hyperferroelectrics.Proper ferroelectrics are those ferroelectric materials in which thephase transition parameter is the electric polarization. Improper andhybrid-improper ferroelectrics are ferroelectric materials in which thephase transition parameter is not the homogeneous electric polarization.These proper versus improper notations refer to the phase transitionitself and the electric polarization in the static ferroelectricmaterial. Hyperferroelectrics are a subclass of proper ferroelectrics inwhich the polarization persists in the presence of a depolarizationfield.

The ferroelectric material (and ferroelectric layer) may besubstantially strain free. By “substantially strain free” it is meantthat the strain in the material is less than about 0.5%, where strain isdefined as the difference between the equilibrium lattice constant ofthe material, a_(f), and the equilibrium lattice constant of theunderlying substrate, a_(s), upon which a layer of the material isgrown, normalized by the lattice constant of the material:(a_(s)−a_(f))/a_(f)×100%. However, in other embodiments, theferroelectric material (and the ferroelectric layer) may be coherentlystrained, e.g., strained SrTiO₃, without relaxation of the latticeconstant of the film. Strain ranges may vary up to +3% (tensile strain)or −3% (compressive strain).

In embodiments, the ferroelectric material of the ferroelectricheterostructure is selected from SrBi₂Ta₂O₉, Bi₄Ti₃O₁₂,PbZr_(1-x)Ti_(x)O₃ (0≤x≤1), BaTiO₃, BiFeO₃, and combinations thereof;and the noncentrosymmetric metal is selected from Ca₃Ru₂O₇, CeCuSn,CeAuSn, CaIrSi₃, CaPtSi₃, LaIrSi₃, LaGeSi₃, LiOsO₃ and combinationsthereof.

In the ferroelectric heterostructure, the ferroelectric layer ischaracterized by exhibiting an electric polarization and the firstelectrode layer comprising the noncentrosymmetric metal is characterizedby exhibiting polar ionic displacements and further wherein, a componentof the polar ionic displacements of the first electrode layer isparallel to a component of the electric polarization of theferroelectric layer. This parallel alignment (versus antiparallel ororthogonal alignment) may be ensured during growth of the ferroelectriclayer/first electrode layer.

The ferroelectric heterostructure may further comprise a secondelectrode layer comprising a second noncentrosymmetric metal, the secondelectrode layer disposed under the ferroelectric layer. Again, theferroelectric layer and the second electrode layer are in direct contactso as to form a ferroelectric-second electrode interface. The result isa sandwich structure (effectively a capacitor) as illustrated in FIG. 3for a ferroelectric heterostructure including a ferroelectric layer ofBaTiO₃ between an upper electrode of the noncentrosymmetric metal LiOsO₃and a lower electrode of the noncentrosymmetric metal LiOsO₃. Inembodiments, such as that shown in FIG. 3, the ferroelectricheterostructure is a symmetric ferroelectric heterostructure, by whichit is meant that the first and second electrode layers are composed ofthe same material(s). However, this is not a requirement. Asymmetricferroelectric heterostructures in which the first and second electrodelayers are composed of different materials may also be used. Selectionof the second noncentrosymmetric metal and the materials of the secondelectrode layer follow that described above with respect to the firstnoncentrosymmetric metal/materials of the first electrode layer.

As noted above, the use of noncentrosymmetric metals means that thecritical thickness value for ferroelectricity in the ferroelectric layereffectively vanishes. This means that the present ferroelectricheterostructures may make use of ferroelectric layers having thicknesseswhich are significantly smaller than those in conventional ferroelectricheterostructures, including thicknesses which are below the criticalthickness value of the selected ferroelectric material(s). As notedabove, a ferroelectric material (and thus the ferroelectric layer) ischaracterized by a critical thickness value, the magnitude of which maybe calculated or measured. In embodiments, the thickness of theferroelectric layer in the ferroelectric heterostructure is less thanthe critical thickness value of the ferroelectric layer. Suchthicknesses would not be used in conventional ferroelectricheterostructures since the heterostructures would be non-functional dueto loss of ferroelectricity. In embodiments, the thickness of theferroelectric layer may be in the range of from about the thickness of asingle unit cell of the selected ferroelectric material to less than thecritical thickness value of the ferroelectric layer. The term“thickness,” when used in reference to the ferroelectric layer in theferroelectric heterostructure (as opposed to its critical thicknessvalue) may refer to an average value as measured from a representativenumber of cross-sections of the ferroelectric layer.

Regarding the first and second electrode layers, the layers may beformed to have a variety of thickness values. Again, the term“thickness” may refer to an average value analogous to that definedabove for the ferroelectric layer. In embodiments, the first and secondelectrode layers may be characterized by an average thickness of atleast about 2 nm, at least about 10 nm, at least about 25 nm, at leastabout 50 nm, at least about 100 nm, etc. Greater average thicknesses mayalso be used.

Even when the ferroelectric heterostructures include ferroelectriclayers having a thickness less than the critical thickness value, theyexhibit ferroelectricity. This may be confirmed using the calculationsdescribed in the Examples, below, e.g., by confirming that the groundstate of the ferroelectric heterostructure is the ferroelectric stateusing energy versus A_(NNO) plots as shown in FIG. 6. Alternatively, theexistence of ferroelectricity may be verified experimentally usingscanning probe microscopies (SPM) or the Positive-Up Negative-Down(PUND) method to obtain the displacement current. (See Journal ofApplied Physics 64, 787 (1988).)

The ferroelectric heterostructures may be used in a variety of devicesand device components, including capacitors (e.g., see FIG. 3) andfield-effect transistors (e.g., see FIGS. 10A-10D).

In embodiments, the first electrode layer of the ferroelectricheterostructure forms only a single ferroelectric-first electrodeinterface, i.e., with the ferroelectric layer. Similarly, in embodimentsincluding the second electrode layer, this second electrode layer mayalso form only a single ferroelectric-second electrode interface, i.e.,with the ferroelectric layer. Other material layers may be in directcontact with the first and second electrode layers, but in suchembodiments, the first and second electrode layers are not in directcontact with another ferroelectric material/layer. In embodiments, theferroelectric layer is the only ferroelectric layer in theheterostructure. Other material layers may be included in theferroelectric structure, but in such embodiments, the heterostructurecomprises a single ferroelectric layer. Such embodiments exclude theferroelectric heterostructures from being a subcomponent of superlatticeor multi-quantum well structures having periodically alternatingferroelectric/noncentrosymmetric metal layers. Thus, such ferroelectricheterostructures are not part of a superlattice or multi-quantum wellstructure.

A variety of known thin film deposition or growth techniques may be usedto form the ferroelectric heterostructures, capacitors, transistors,etc.

Illustrative applications for the present disclosure include thefollowing: ferroelectric capacitors to make ferroelectric random accessmemory (FeRAM) for computer and electronic devices; ferroelectric tunneljunctions (FTJs) utilizing ferroelectric capacitor structures andassociated solid state memories based on them; ferroelectricfield-effect transistors (FeFET); integration into high dielectricrandom access memory (DRAM); ferroelectric capacitors forradio-frequency identification technology; and ferroelectric capacitorsfor sensor applications such as medical ultrasound machines, highquality infrared cameras, fire sensors, sonar, vibration sensors, andeven fuel injectors on diesel engines.

EXAMPLES

Introduction

When ferroelectric (FE) oxides are utilized in metal/oxideheterostructures and nanocapacitors, scaling of the active FE isrequired to improve performance [1-3]. Nonetheless, deleteriousnanoscale effects are amplified in these geometries [4-6] and act toeliminate the functional electric polarization [7-9]. The loss offerroelectricity in nanoscale capacitors frequently occurs when thepolarization is perpendicular to the film surface [4, 10-12], i.e., thedesired polarization direction for field-tunable devices [13]. Boundcharges are only partially screened at the interface, resulting in astrong depolarizing field that suppresses the polarization. Indeed,numerous reports suggest a critical thickness, t_(FE)*, below which theelectric polarization disappears. Experimental studies on PbTiO₃ findt_(FE)*˜20 Å at 300 K [14], whereas t_(FE)*˜4.0 nm, inPb(Zr_(0.2)Ti_(0.8))O₃ films [8]. Furthermore, first-principles densityfunctional theory (DFT) calculations predict t_(FE)*˜2.4 nm insingle-domain BaTiO₃ films between SrRuO₃ electrodes [4], which reducesto t_(FE)*˜1.0 nm after accounting for ionic relaxation in theelectrodes [5, 6]. With this limitation, ferroelectric based devices areunable to meet the continuous scaling changes demanded by higher densitydata storage technologies.

Although proposals to overcome the problem exist, a general solutionremains elusive. Most approaches focus on tuning the stability of the FEstate. Epitaxial strain engineering has been proposed; nonetheless, thisstrategy extends to a limited number of oxides, requires complexprocessing steps, and is limited by available commercial substrates. Forexample, although it is predicted that t_(FE)*→0 in thin films of theincipient ferroelectric BaZrO₃, a large epitaxial compressive strain of4.25% is required [15], which would produce deleterious misfitdislocations. Integration with silicon is likely to also lead touncontrolled interface states [16]. Furthermore, at this level of straint_(FE)* is still finite for BaTiO₃ (Ref. 15).

Alternative solutions change the type of ferroelectric and the activeinversion symmetry lifting mechanism. Ab-initio calculations find thatt_(FE)*→0 using so-called hyperferroelectrics [17], which have apersistent polarization different from proper FE oxides, or improperferroelectrics [18, 19]. However, hyperferroelectric bulk materials (andthin films) remain to be synthesized [20]. Another route relies oncreating an enhanced interfacial FE state by controlling the covalencyof the metal-oxygen bond at the heterointerface [21-24].

Here the critical thickness is examined for ferroelectricity innanocapacitors consisting of polar metal electrodes and conventionalferroelectric oxides under short-circuit boundary conditions using DFTcalculations. Recently, polar metals have garnered considerable interest[25, 26] because they exhibit simultaneously inversion-liftingdisplacements and metallicity. In these compounds, the polardisplacements are weakly coupled to the states at the Fermi level, whichmakes possible the coexistence of a polar structure and metallicity[27]. The main finding is that polar metal electrodes suppress thecritical thickness limit through interfacial polar displacements, whichstabilize the ferroelectric (polarized) state; this geometric effectdoes not rely on interfacial bond chemistry or ‘perfect’ screening ofthe depolarization field, but rather results from the intrinsic brokenparity present in the electrode.

Methods

First-principles DFT calculations were performed within thelocal-density approximation (LDA) and hybrid functional (HSE06, Ref. 28and 29) as implemented in the Vienna Ab initio Simulation Package (VASP)[30] with the projector augmented wave (PAW) approach [31] to treat thecore and valence electrons using the following electronicconfigurations: 1s²2s² (Li), 5p⁶6s²5d⁶ (Os), 2s²2p⁴ (0), 5s²5p⁶6s² (Ba),3d²4s² (Ti), 4s²4p⁶5s² (Sr), 4d⁷5s¹ (Ru), 2p⁶3s¹ (Na), 4p⁶4d⁴5s¹ (Nb).The Brillouin zone integrations are performed with a 13×13×1Monkhorst-Pack k-point mesh [32] and a 600 eV plane wave cutoff for theLiOsO₃/NaNbO₃/LiOsO₃ and SrRuO₃/BaTiO₃/SrRuO₃ capacitor structures. Theatomic positions (force tolerance less than 0.1 meV A°−1) were relaxedusing Gaussian smearing (20 meV width).

Below 150 K, NaNbO₃ and LiOsO₃ are isostructural with rhombohedral spacegroup R3c and pseudocubic lattice parameters of 3.907 Å (Ref. 25), (Ref33) and 3.650 Å, respectively. Owing to the large lattice mismatchbetween the two compounds, a symmetric ferroelectric capacitor structurewas simulated with an LiO/NbO₂ interfacial termination, shown in FIG.4A, under an epitaxial constraint that would be imposed by a(La_(0.3)Sr_(0.7))(Al_(0.65)Ta_(0.35))O₃ substrate [34]. Theout-of-plane lattice parameter was also relaxed. This resulted in acompressive strain of ˜1% for NaNbO₃ and a tensile strain of ˜6% forLiOsO₃. Note that at the bulk level, it was found that a tensile strainof about greater than 6% suppressed the polar instability along the[001]-pseudocubic direction of LiOsO₃. Moreover, NaNbO₃ was selected forthe capacitor structures in order to eliminate any charge transfer dueto ‘polar catastrophe/charge mismatch’ physics as the interface:[LiO]¹⁻, [NaO]¹⁻, [NbO₂]¹⁺, and [OsO₂]¹⁺.

For the two ferroelectric capacitors, the layered-oxide notation used inRef 4 was adopted, that is:

-   -   [LiO—(OsO₂—LiO)_(n)/NbO₂—(NaO—NbO₂)_(m)] and    -   [SrO—(RuO₂—SrO)_(n)/TiO₂—(BaO—TiO₂)_(m)]        to clearly demarcate the interface composition in the        LiOsO₃/NaNbO₃/LiOsO₃ and SrRuO₃/BaTiO₃/SrRuO₃ capacitors,        respectively. A LiO/NbO₂ electrode/ferroelectric interface for        the LiOsO₃/NaNbO₃/LiOsO₃ capacitor and a SrO/TiO₂ interface        termination for the SrRuO₃/BaTiO₃/SrRuO₃ capacitor were used.        For both ferroelectric capacitors, the number of 5-atom        perovskite units of the electrode at n=6 was constrained to        ensure a thickness large enough to avoid interaction between the        two interfaces, and m ranged from 1 to 3. The periodic boundary        conditions naturally impose the required short-circuit condition        between the electrodes. Note that for SrRuO₃/BaTiO₃/SrRuO₃        capacitors, the geometry differs only slightly from that used by        Junquera and Ghosez [4], whereby the thickness of this electrode        is greater.

The group theoretical analysis was aided by the ISODISTORT software[35]. This software was used to evaluate the geometric-induced inversionsymmetry-breaking displacements of the P4mm structure with respect theP4/mmm phase, reducing the polar structure into a set ofsymmetry-adapted modes associated with different irreduciblerepresentations of the P4/mmm phase. The “robust” algorithm was used tomatch an atom in the undistorted structure to every atom in thedistorted structure separated by a threshold distance less than 3 Å.

Results and Discussion

The first ferroelectric nanocapacitor that was focused on consisted offerroelectric NaNbO₃ of varying thickness m confined between electrodesof the experimentally known polar metal LiOsO₃ (see FIG. 4A, m=2) [25,36]. The layered-oxide notation used in Ref 4 was adopted, that is[LiO—(OsO₂—LiO)_(n)/NbO₂—(NaO—NbO₂)_(m)], to clearly demarcate theinterface composition (see Methods). Two symmetric nanocapacitors werecreated with a polar and paraelectric configuration for both LiOsO₃ andNaNbO₃, respectively. The out-of-plane lattice parameter and the atomicspositions of the nanocapacitors for m=1, 2, and 3 were then relaxed. Thelowest energy heterostructures were polar with space group P4mm andexhibit large Li ions displacements along the [001]-pseudocubicdirection (FIG. 4B). No zone-center dynamical instabilities were foundin these heterostructures. Note that structures with an initialparaelectric configuration relaxed into a centrosymmetric structure(space group P4/mmm) with Li atoms displaying large antipolardisplacements in LiOsO₃ (not shown) that decrease towards the interface(FIG. 4C).

Representation theory analysis was used to examine the inversion liftingdistortions (see Methods), and it was found that the distortion vectorcorresponds to the irreps Γ₁ ⁺ and Γ₃ ⁻. The irrep Γ₁ ⁺ reduced theantipolar displacements in LiOsO₃, resulting in the centrosymmetricP4/mmm structure depicted in FIG. 4A. The irrep Γ₃ ⁻ differs in that itis a polar mode which involves mainly Li ion displacements—the maximumamplitudes being ˜1.3 Å in LiOsO₃, with decreasing amplitude towards theLiO/NbO₂ interface (FIG. 4B). It also consists of polar displacements ofall ions in the dielectric NaNbO₃ layers with the Nb ions off-centeringthe most (FIG. 4D)

For all thicknesses, the NaNbO₃ thin film maintained a ferroelectricground state characterized by both Nb and Na displacements (FIG. 4B). Alinear polarization-displacement model using the Born effective chargefrom Ref. 37 resulted in a 0.86 C m⁻² polarization for NaNbO₃ in m=2. Ananalysis of the differential ionic relaxations in the heterostructurerevealed polar displacements at the LiO/NbO₂ interfaces—an interfacialferroelectricity—which are a consequence of the polar metal used as anelectrode. This produced an enhanced polarization in the ferroelectriccompared to that calculated using the aforementioned procedure in theexperimental R3c structure (0.59 Cm⁻²) [38]. In particular, although thetwo interfaces of the paraelectric structures exhibited antiparallelpolar displacements (FIG. 4C) while the interfaces of the polar groundstate structures had parallel polar displacements, as shown in FIG. 4D.

FIGS. 5A-5B show the electronic properties for the NaNbO₃ layer in theLiOsO₃/NaNbO₃/LiOsO₃ (m=1) nanocapacitor. The LDA functional predictsthat the NaNbO₃ film will be metallic, rendering the ferroelectriccapacitor unusable (FIG. 5A). This behavior is artificial, and is due tothe tendency of the LDA functional to underestimate the band gap ininsulating compound [39, 40]. This pathological problem was solved forDFT by using a more sophisticated functional which includes a fractionof exact exchange (HSE06). FIG. 5B shows that the hybrid functionalfully opened the gap between the O 2p and Nb 4d states. Moreover, it wasfound that the HSE06-relaxed structures exhibited displacements similarto those obtained from the LDA functional; importantly, polardisplacements at the LiO/NbO₂ interfaces. This result supports theconclusion that the interfacial ferroelectricity was induced by thepolar crystal structure of the metallic electrode and not due tospurious shorting of the capacitor. In the remainder of this paper,results obtained using LDA owing to the similar crystal structureobtained with HSE06 functional are reported.

FIG. 6 shows the evolution of the total energy of each capacitor withmode amplitude A_(NNO), which describes the atomic displacementsinvolved in the soft mode of the NaNbO₃ film. The largest energy gainoccurred when the thickness of the ferroelectric film increased. Notethat the shape of the energy surface did not exhibit the characteristicdouble well behavior, because in the calculations the polardisplacements were fixed in the metallic electrodes and only theamplitude of the polar displacements in NaNbO₃ was changed. Independentof the NaNbO₃ film thickness, the energy was minimized for theferroelectric ground state (A_(NNO)≠0), indicating that an idealferroelectric capacitor can be reduced to an ultrathin (single unitcell) size, i.e., t_(FE)*→0. The disappearance of the critical-thicknesslimit to ferroelectricity was the result of the parallel polardisplacements present at the electrode/dielectric interfaces (FIG. 4B).The ferroelectric state in ultrathin-film devices depends crucially onthe nature of the chemical bonds at the metal/oxide interface. Here,this interfacial bonding occurred and was an immediate consequence ofthe structure of the polar-metal electrodes. The enhanced and parallelinterfacial polar displacements “imprinted” and lead to an overallenhancement of the ferroelectric instability of the film, which isassessed further below. It is emphasized that the interfacial dipoledistortions were due to a geometrical mechanism driven by the polarstructure of the metallic electrode and not due to the stiffness of theelectrode-oxidebonds.

A SrRuO₃/BaTiO₃/SrRuO₃ capacitor (FIG. 7A) was next examined, which asbefore may be written as [SrO—(RuO₂—SrO)_(n)/TiO₂—(BaO—TiO₂)_(m)] toreveal the layered monoxide planes in the structures. The SrO/TiO₂interface geometry was focused on to demonstrate the generality of thissolution to the critical thickness problem. The in-plane latticeparameters were constrained to that of SrTiO₃ (3.905 Å), and theout-of-plane lattice parameter and the atomic positions were relaxed(see Methods), examining capacitors with m=1, 2, and 3 that were wellbelow the reported m=7 critical thickness [4]. With nonpolar SrRuO₃electrodes, the paraelectric configuration is energetically more stablethan the ferroelectric configuration for all BaTiO₃ film thicknessesstudied. This is confirmed by the increase in total energy as a functionof the polar mode amplitude A_(BTO) (FIG. 7B). Indeed, the use of acentrosymmetric metal for the thinnest ferroelectric film resulted in anantisymmetric poling effect of the two interfaces, which forbid thepossibility of a ferroelectric displacement [5].

A computational experiment was then performed whereby centric SrRuO₃ wastransmuted into a hypothetical polar metal by following the design rulesfor noncentrosymmetric metals introduced in Ref. 27. This was done byimposing a polar distortion in SrRuO₃, which involved only the Sr atoms,as the orbital character at the Fermi level has a negligiblecontribution from these atoms, with parallel polar displacements at theSrO/TiO₂ interfaces as suggested by the LiOsO₃/NaNbO₃/LiOsO₃ capacitorresults. Bulk SrRuO₃ does not exhibit polar distortions, and here it wasmade artificially polar to isolate the interfacial geometric effectindependent of chemistry with respect to the model with centrosymmetricelectrodes.

In FIG. 8A the energy evolution of this hypothetical capacitor as afunction of the mode amplitude A_(BTO) is shown, with parallel polar Srdisplacements imposed uniformly at 0.07 Å with respect to thecentrosymmetric structure (see inset). In contrast to FIG. 7B, theferroelectric state of BaTiO₃ was more stable than the paraelectricgeometry for all thicknesses m=1, 2, and 3, as indicated by the energygain for A_(BTO)≠0. These results indicate that t_(FE)*→0 in the BaTiO₃film between the polar-metal SrRuO₃.

The energy gain was strongly influenced by the polar displacements ofthe Sr atoms. In particular, by doubling the amplitude of the polardisplacements of the Sr atoms, from 0.07 Å to 0.14 Å for the case m=1,it was found that the energy gain increased from μ3 meV to ˜10 meV (FIG.8B). Comparing FIGS. 8A and 8B, a shift was found in the critical modeamplitude to larger values, which suggests that the device containing apolar-metal electrode with larger Sr displacements displayed a largerferroelectric polarization. Indeed, when the other limit was consideredby fully removing the polar displacements at the SrO/TiO₂ interface(setting them to 0 Å), the energy landscape presented in FIG. 7B wasrestored. Note that for the disappearance of the critical thickness, itwas necessary that the polar direction (or a component of it) in theelectrodes, and therefore that of the interfacial dipole, coincided withthe direction of polarization of the ferroelectric film (FIG. 8B,inset).

It could be argued that the findings here were a result of the polarmetals better screening the ferroelectrics polarization; however, thiswas not the case. Indeed, the electrostatic for the BaTiO₃ nanocapacitorwith “polar” and nonpolar SrRuO₃ electrodes were almost the same (FIG.9). Moreover, polar metals typically have longer screening lengths thanconventional metals [41]. This result further confirms the role ofinterfacial geometric effects induced by the polar structure of themetallic electrode in controlling the critical thickness.

Lastly, how the proposed device can be switched is discussed. Polarmetals are not ferroelectrics. Indeed, the application of an electricfield cannot switch the polar distortion in the metal because the freeelectrons will screen the electric field. However, it has been shownthat the polar distortion in the metal can be switched by applying anelectric field to a superlattice composed of an insulating ferroelectricmaterial and a polar metal by coupling to the ferroelectric polarization[42]. Similarly, when an electric field is applied to the aforementionednanocapacitors, the polar distortion in the ferroelectric thin filmshould align along the direction of the electric field; then, because ofthe interfacial coupling between the polar metallic electrode and theferroelectric film [42], the polar displacements in the polar metal andthe interfacial polar displacements should follow. Note that thegeometric configuration required to sustain ferroelectricity, i.e.,t_(FE)*→0, is preserved in the switching mechanism. Alternativeapproaches have also been applied to degenerately dopedferroelectrics.[43]

CONCLUSION

In summary, a ferroelectric capacitor is proposed wherein theconventional metallic electrodes are replaced by noncentrosymmetricmetallic electrodes. It was shown that the polar displacements in thenoncentrosymmetric metallic electrodes induced interfacialferroelectricity, which supported a polar instability in theferroelectric film regardless of the dielectric thickness. AlthoughLiOsO₃ was utilized herein for simplicity, these results are general andthe same conclusions may be achieved using other noncentrosymmetricmetals as electrodes with the described geometric constraints (see Ref.44 for a list of materials). These polar-metal based nanoscalecapacitors maintain the functionality of the ferroelectric filmindependent of the degree of miniaturization and lead to devicearchitectures with improved scalability.

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The word “illustrative” is used herein to mean serving as an example,instance, or illustration. Any aspect or design described herein as“illustrative” is not necessarily to be construed as preferred oradvantageous over other aspects or designs. Further, for the purposes ofthis disclosure and unless otherwise specified, “a” or “an” means “oneor more.”

Use of directional terms such as “under” and “over” is not meant to belimiting, but rather to establish relative relationships betweenelements and points of reference.

The foregoing description of illustrative embodiments of the inventionhas been presented for purposes of illustration and of description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of theinvention. The embodiments were chosen and described in order to explainthe principles of the invention and as practical applications of theinvention to enable one skilled in the art to utilize the invention invarious embodiments and with various modifications as suited to theparticular use contemplated. It is intended that the scope of theinvention be defined by the claims appended hereto and theirequivalents.

What is claimed is:
 1. A ferroelectric heterostructure comprising aferroelectric layer comprising a ferroelectric material and a firstelectrode layer comprising a first noncentrosymmetric metal, the firstelectrode layer disposed on the ferroelectric layer to form aferroelectric-first electrode interface, wherein the ferroelectric layeris characterized by exhibiting an electric polarization and the firstelectrode layer is characterized by exhibiting polar ionic displacementsand further wherein, a component of the polar ionic displacements of thefirst electrode layer is parallel to a component of the electricpolarization of the ferroelectric layer.
 2. The heterostructure of claim1, wherein the ferroelectric material is a ferroelectric oxide.
 3. Theheterostructure of claim 2, wherein the ferroelectric oxide is selectedfrom SrBi₂Ta₂O₉, Bi₄Ti₃O₁₂, PbZr_(1-x)Ti_(x)O₃ (0≤x≤1), BaTiO₃, BiFeO₃,and combinations thereof.
 4. The heterostructure of claim 1, wherein theferroelectric material is selected from BaTiO₃; strained SrTiO₃;Ba_(x)Sr_(1-x)TiO₃ wherein 1>x>0; Bi₄Ti₃O₁₂; PbTiO₃; BaZrO₃;Pb(Zr_(x)Ti_(1-x))O₃ wherein 1≥x≥0; (Sr,Ba)Nb₂O₆; NaNbO₃; BiFeO₃; YMnO₃;Sr_(x)Ca_(3-x)Ti₂O₇ wherein 1>x>0; SrBi₂Ta₂O₉; LiNbO₃; Sr(Ta,Nb)₂O₇;Gd₂(MoO₄)₃; Pb₅Ge₃O₁₁; BaMnF₄; GeTe; SrAlF₅; SbSI; and combinationsthereof.
 5. The heterostructure of claim 1, wherein the ferroelectriclayer is substantially strain free.
 6. The heterostructure of claim 1,wherein the first noncentrosymmetric metal is selected from Rh₂Ga₉;Ru₇B₃; BiPd; UIr; Mg₂Al₃; Ir₉Al₂₈; Ir₂Ga₉; Rh₂Ga₉; γ-Bi₂Pt;Au_(6.05)Zn_(12.51); Ba₂₁Al₄₀; Cr₅Al₈; Mn₅Al₈; Cu_(7.8)Al₅; Cu₇Hg₆;NbS₂; Sn₄As₃; Sn₄P₃; REPt₃B, wherein RE is selected from La, Pr, and Nd;La₅B₂C₆; LaNiC₂; La₂NiAl₇; CaAlSi; Li₂IrSi₃; CeRuSi₃; SrAuSi₃; CePt₃Si;AIrSi₃, wherein A is selected from Ca and Ce; ARhSi₃, wherein A isselected from Ce and La; APtSi₃, wherein A is selected from Ca, Ba, andEu; ACoGe₃, wherein A is selected from Ce, Pr, and La; LaASi₃, wherein Ais selected from Ir and Pd; CeAGe₃, wherein A is selected from Ir andRh; EuNiGe₃; EuPdGe₃; LaAGe₃, wherein A is selected from Co, Fe, Ir andRh; SrAGe₃, wherein A is selected from Pd and Pt; REPdIn₂, wherein RE isselected from Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu and Y; REAuGe,wherein RE is selected from Ce, Lu, Sc and Ho; La₁₅Ge₉X, wherein X isselected from C, Co, Fe, and Ni; HoAuGe; CeCuSn; Sr₃Cu₈Sn₄; Li₁₇Ag₃Sn₆;LiOsO₃; Ca₃Ru₃O₇; RESr₂Cu₂GaO₇, wherein RE is any of La through Yb or Y;BaVS₃; AV₆S₈, wherein A is selected from K, Rb, Cs, and Tl; La₄Mg₅Ge₆;La₄Mg₇Ge₆; Yb₂Ga₄Ge₆; ErPdBi; LuPtBi; Ce₂Rh₃(Pb, Bi)₅; Eu₂Pt₃Sn₅;Lu₄Zn₅Ge₆; IrMg_(2.03)In_(0.97); IrMg_(2.20)In_(0.80); CeAuSn; LaGeSi₃;and combinations thereof.
 7. The heterostructure of claim 1, wherein thefirst noncentrosymmetric metal is selected from Ca₃Ru₂O₇, CeCuSn,CeAuSn, CaIrSi₃, CaPtSi₃, LaIrSi₃, LaGeSi₃, LiOsO₃ and combinationsthereof.
 8. The heterostructure of claim 7, wherein the ferroelectricmaterial is a ferroelectric oxide.
 9. The heterostructure of claim 8,wherein the ferroelectric oxide is selected from SrBi₂Ta₂O₉, Bi₄Ti₃O₁₂,PbZr_(1-x)Ti_(x)O₃ (0≤x≤1), BaTiO₃, BiFeO₃, and combinations thereof.10. The heterostructure of claim 9, wherein the ferroelectric oxide isBaTiO₃ and the first noncentrosymmetric metal is LiOsO₃.
 11. Theheterostructure of claim 1, further comprising a second electrode layercomprising a second noncentrosymmetric metal, wherein the ferroelectriclayer is between the first and second electrode layers to further form aferroelectric-second electrode interface, wherein the second electrodelayer is characterized by exhibiting polar ionic displacements andfurther wherein, a component of the polar ionic displacements of thesecond electrode layer is parallel to the component of the electricpolarization of the ferroelectric layer.
 12. The heterostructure ofclaim 11, wherein the first and second noncentrosymmetric metals are thesame such that the heterostructure is a symmetric heterostructure. 13.The heterostructure of claim 1, wherein the ferroelectric layer is hasan average thickness which is less than a critical thickness value ofthe ferroelectric layer.
 14. The heterostructure of claim 13, whereinthe average thickness is in the range of from about the thickness of asingle unit cell of the ferroelectric material to less than the criticalthickness value.
 15. A field-effect transistor comprising theheterostructure of claim 1 disposed over a substrate, a sourceelectrically coupled to the substrate and a drain electrically coupledto the substrate.
 16. The transistor of claim 15, further comprising abuffer layer between the heterostructure and the substrate.
 17. Afield-effect transistor comprising the heterostructure of claim 11disposed over a substrate, a source electrically coupled to thesubstrate and a drain electrically coupled to the substrate.
 18. Thetransistor of claim 17, further comprising a buffer layer between theheterostructure and the substrate.
 19. The transistor of claim 17,wherein the substrate is a perovskite substrate.
 20. A method of usingthe heterostructure of claim 1, the method comprising applying anelectric field across the heterostructure.